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A Minimal Biophysical Model of Neocortical Pyramidal Cells: 1
Implications for Frontal Cortex Microcircuitry and Field Potential 2
Generation 3
Beatriz Herrera1, Amirsaman Sajad2, Geoffrey F. Woodman2, Jeffrey D. Schall2, Jorge J. 4
Riera1* 5
1 Department of Biomedical Engineering, Florida International University, Miami, Florida 6
33174, USA; 2Department of Psychology, Vanderbilt Vision Research Center, Center for 7
Integrative & Cognitive Neuroscience, Vanderbilt University, Nashville, Tennessee 37203, USA 8
*Corresponding author 9
E-mail: [email protected] (JR) 10
Number of pages: 38 11
Number of figures: 6 12
Number of tables: 3 13
Number of words for abstract, introduction, and discussion: 234, 641, and 1444. 14
Conflicts of interest: None 15
Acknowledgments 16
This work was supported by FIU SEED – grant Wallace Coulter Foundation (BH, JR), 17
R01-EY019882 (JDS, GFW, BH, JR), and by Robin and Richard Patton through the E. Bronson 18
Ingram Chair in Neuroscience (JDS). The authors thank Michelle Schall and Arash Moshkforoush 19
for helpful discussions and comments on the manuscript. 20
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Abstract 21
Ca2+ spikes initiated in the apical dendrites of layer-5 pyramidal cells (PC) underlie 22
nonlinear dynamic changes in the gain of cellular response, which is critical for top-down cognitive 23
control. Detailed models with several compartments and dozens of ionic channels have been 24
proposed to account for this Ca2+ spike-dependent gain with its associated critical frequency. 25
However, current models do not account for all known Ca2+-dependent features. Previous attempts 26
to include more features have required increasing complexity, limiting their interpretability and 27
utility for studying large population dynamics. We present a minimal 2-compartment biophysical 28
model, overcoming these limitations. In our model, a basal-dendritic/somatic compartment 29
included typical Na+ and K+ conductances, while an apical-dendritic/trunk compartment included 30
persistent Na+, hyperpolarization-activated cation (Ih), slow inactivation K+, muscarinic K+, and 31
Ca2+ L-type. The model replicated the Ca2+ spike morphology and its critical frequency plus three 32
other defining features of layer-5 PC synaptic integration: linear frequency-current relationships, 33
backpropagation-activated Ca2+ spike firing, and a shift in the critical frequency by blocking Ih. 34
Simulating 1,000 synchronized layer-5 PCs, we reproduced the current source density patterns 35
evoked by Ca2+-spikes both with and without Ih current. Thus, a 2-compartment model with five 36
non-classic ionic currents in the apical-dendrites reproduces all features of these neurons. We 37
discuss the utility of this minimal model to study the microcircuitry of agranular areas of the frontal 38
lobe involved in cognitive control and responsible for event-related potentials such as the error-39
related negativity. 40
Key words 41
pyramidal cells; LFP sources; cortical microcircuitry; biophysical modeling; cognitive 42
control 43
44
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Significance Statement 45
A tractable model of layer-5 pyramidal cells replicates all known features crucial for distal 46
synaptic integration in these neurons. This minimal model enables new multi-scale investigations 47
of microcircuit functions with associated current flows measured by intracranial local field 48
potentials. It thus establishes a foundation for the future computational evaluation of scalp 49
electroencephalogram signatures imprinted by Ca2+ spikes in pyramidal cells, a phenomenon 50
underlying many brain cognitive processes. 51
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Introduction 52
Models of the neocortical microcircuit with biophysically plausible pyramidal cells (PC) 53
are necessary to translate between observed patterns of neural spiking, local field potentials (LFP), 54
and the derived scalp electroencephalogram (EEG). Layer 5 (L5) PCs have an elongated 55
morphology with dendrites spanning all cortical layers; hence their synaptic activity causes 56
laminar current sources (Einevoll et al., 2013; Reimann et al., 2013). The local synchronization of 57
a large population of L5-PCs produces electric potentials that can be measured on the scalp 58
(Hämäläinen et al., 1993; Riera et al., 2012). Integrative features of L5-PCs have suggested their 59
participation in signaling coincident inputs to basal-dendritic/somatic and apical-dendritic regions 60
(Larkum, 2013). The arrival times of sensory inputs, efferent copies, and task rules in agranular 61
frontal cortex are critical in cognitive control (Sajad et al., 2019; Subramanian et al., 2019). One 62
well-characterized cognitive control function is error monitoring by the medial frontal cortex 63
(Stuphorn et al., 2000; Sajad et al., 2019), which is indexed by an error-related negativity (ERN) 64
in scalp potentials (Gehring et al., 1993). Therefore, models of L5-PCs will help clarify the 65
electrogenesis of the ERN. 66
L5-PCs exhibit two distal excitability zones, endowing these neurons with important 67
integrative features. One excitability zone, at the axon hillock, produces typical Na+ action 68
potentials (AP) and another, in the distal trunk, produces Ca2+-spikes (Amitai et al., 1993; Yuste 69
et al., 1994; Schiller et al., 1997; Larkum and Zhu, 2002). The coincidence of a Na+-AP with an 70
apical dendritic excitatory postsynaptic potential produces additional APs via a backpropagation-71
activated Ca2+-spike, “BAC” firing (Larkum et al., 1999b). Na+-APs show a linear frequency-72
current (f-I) relation with different sensitivities at the two excitability zones (Larkum et al., 2004). 73
Dendritic Ca2+-spikes generated by strong inputs show a sustained depolarization (Larkum et al., 74
2001) that produces high-frequency Na+-APs (Schwindt and Crill, 1999; Williams and Stuart, 75
1999; Larkum et al., 2001). L5-PCs exhibit a critical frequency (CF) between 60 and 200 Hz for 76
eliciting Ca2+-spikes (Larkum et al., 1999a) via somatic stimulation, which is sensitive to the 77
hyperpolarization-activated cation current, Ih, in apical-dendrites (Berger et al., 2001). 78
Previously proposed biophysical models with 2 or 3 neuronal compartments and fewer 79
conductances explained only isolated features (i.e., the I-f curves - Larkum et al., 2004; the BAC-80
firing - Chua et al., 2015; and Yi et al., 2017). Biophysical models of higher complexity accounted 81
for some combinations of the three major features: the BAC-firing (Rapp et al., 1996; Schaefer et 82
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5
al., 2003; Hay et al., 2011; Bahl et al., 2012; Almog and Korngreen, 2014; Mäki-Marttunen et al., 83
2018), the f-I curves (Hay et al., 2011; Bahl et al., 2012; Mäki-Marttunen et al., 2018), and the CF 84
of Ca2+-spikes (Schaefer et al., 2003; Hay et al., 2011; Bahl et al., 2012; Almog and Korngreen, 85
2014). However, single cell models with many compartments and ionic channels are 86
computationally expensive to use in large-scale simulations of neocortical networks. Furthermore, 87
fitting these complex models to LFP/EEG data is practically unattainable, limiting interpretability 88
and their applications to other research areas. Only one previous model replicated realistic [Ca2+] 89
dynamics in the distal-trunk during Ca2+-spikes (Mäki-Marttunen et al., 2018). Furthermore, no 90
previous model has accounted for the Ih shift of CF, the current source density (CSD) patterns 91
associated with dendritic Ca2+-spikes evoked by somatic stimulation of PCs above the CF, and the 92
effect of blocking Ih on these patterns (Suzuki and Larkum, 2017). 93
We describe the simplest possible biophysical model (2-compartments, 7 ionic 94
conductances) of L5-PCs accounting for all these features. In particular, it reproduced Ca2+ 95
dynamics above the CF and explained the shift produced by Ih. The model replicates CSD patterns 96
obtained from synchronized Ca2+-spikes of 1,000 L5-PCs evoked by supra-CF somatic 97
stimulation. Therefore, this minimal L5-PC model will be crucial for the interpretation of LFP-98
CSD/EEG patterns associated with cognitive control based on our understanding of the agranular 99
laminar microcircuitry (Sajad et al., 2019). 100
Materials and Methods 101
L5-PC minimal model 102
We modeled the L5-PC as a 2-compartment neuron with a compartment representing the 103
basal-dendrites/soma, and another compartment representing its distal-trunk (Ca2+-spike initiation 104
zone) and the apical-dendrites. The trunk is represented by a transfer resistance (𝑅𝑇) between the 105
two compartments (Figure 1A). The basal-dendritic/somatic compartment includes the classic 106
Hodgkin-Huxley sodium (𝐼𝑁𝑎) and potassium delayed rectifier (𝐼𝐾𝑑𝑟) currents (Hodgkin and 107
Huxley, 1952). The apical-dendrite/trunk compartment includes persistent 𝑁𝑎+ current (𝐼𝑁𝑎𝑝) 108
(Magistretti and Alonso, 1999), 𝐶𝑎2+ L-type current (𝐼𝐶𝑎𝐿) (Lytton and Sejnowski, 1991), 109
hyperpolarization-activated non-specific cation current (𝐼ℎ) (Kole et al., 2006), muscarinic K+ 110
current (𝐼𝑀) (Adams et al., 1982), and the slow-inactivating potassium current (𝐼𝐾𝑠) (Korngreen 111
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and Sakmann, 2000). The membrane potentials of the two compartments are given by the 112
following coupled differential equations: 113
𝐶𝑚𝑠
𝑑𝑉𝑠
𝑑𝑡= −𝐼𝑁𝑎 − 𝐼𝐾𝑑𝑟 − 𝐼𝑙
𝑠 +(𝑉𝑑 − 𝑉𝑠)
𝑅𝑇+ 𝐼𝑖𝑛𝑗
𝑠 (1)
𝐶𝑚𝑑
𝑑𝑉𝑑
𝑑𝑡= −𝐼𝑁𝑎𝑝 − 𝐼𝐶𝑎𝐿 − Iℎ − IM − 𝐼𝐾𝑠 − 𝐼𝑙
𝑑 +(𝑉𝑠 − 𝑉𝑑)
𝑅𝑇+ 𝐼𝑖𝑛𝑗
𝑑 (2)
where subscripts s and d denote the basal-dendritic/somatic and apical-dendritic/trunk 114
compartments, respectively. 𝑉𝑖, 𝐶𝑚𝑖 , 𝐼𝑙
𝑖 and 𝐼𝑖𝑛𝑗𝑖 (𝑖 ∈ {𝑠, 𝑑}) represent the membrane potential, 115
membrane capacitance, leak current and injected current for the i-th compartment, respectively 116
(Table 1 – parameters). The ionic currents are modeled using the Hodgkin-Huxley formalism in 117
which: 118
𝐼𝑘 = 𝑔𝑘𝑚𝑘𝑥ℎ𝑘
𝑦(𝑉𝑖 − 𝐸𝑘) (3)
where, 𝑔𝑘 is the maximal conductance of the k-th ionic channel; 𝑚𝑘 and ℎ𝑘 are its 119
activation and inactivation gating variables (Table 2 – ionic current kinetics); 𝑥 and 𝑦 are their 120
respective exponents; and 𝐸𝑘 is the equilibrium potential of the k-th ion. The leak current was 121
modeled by 𝐼𝑙𝑖 = 𝑔𝑙
𝑖(𝑉𝑖 − 𝐸𝑙𝑖). All the equilibrium potentials are considered constant, except for 122
the equilibrium potential of 𝐶𝑎2+, which depends on the intracellular 𝐶𝑎2+ concentration 123
([𝐶𝑎2+]𝑖) through the Nernst equation. Because of ionic diffusion, we treat [Ca2+] as a stochastic 124
variable. Therefore, we added a Wiener noise 𝑔𝐶𝑎𝑑𝑊𝐶𝑎 to equation (4) using the approach 125
described in a previous study (Riera et al., 2011), with 𝑔𝐶𝑎 = 1 × 10−9. 126
The intracellular 𝐶𝑎2+concentration dynamics is given by 127
𝑑[𝐶𝑎2+]𝑖
𝑑𝑡= −
𝛾𝑘(𝐼𝐶𝑎𝐿(𝑉𝑑) − 𝐼𝐶𝑎𝐿(𝑉𝑑𝑟))
2Fd−
([𝐶𝑎2+]𝑖 − [𝐶𝑎2+]𝑖𝑟)
𝜏𝑅 (4)
where 𝑉𝑑𝑟 is the dendritic resting potential, [𝐶𝑎2+]𝑖
𝑟 is the intracellular 𝐶𝑎2+ concentration 128
at rest, 𝜏𝑅 = 80𝑚𝑠 is the decay time constant of the intracellular 𝐶𝑎2+ concentration due to active 129
transport (Schaefer et al., 2003). 𝑑 = 1 𝜇𝑚 is the depth of the submembrane Ca2+ shell, 𝐹 =130
96489 𝐶/𝑚𝑜𝑙 is the Faraday’s constant, and 𝑘 = 10,000/𝐴𝑑 is the unit conversion constant for 131
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𝐼𝐶𝑎𝐿 (mA). The surface area 𝐴𝑑 = 9302.3 𝜇𝑚2 of the apical-dendrite/trunk compartment was 132
calculated based on the values given by Larkum et al. (2004) for parameters 𝐶𝑚𝑑 and 𝑅𝑚
𝑑 . 𝛾 133
represents the fraction of free Ca2+ (not buffered), which was adjusted to reproduce experimental 134
data for [𝐶𝑎2+]𝑖 in the distal-trunk (Larkum et al., 1999a). The basal intracellular Ca2+ was set at 135
its typical physiological value [𝐶𝑎2+]𝑖𝑟 = 80 𝑛𝑀. 136
*** Please insert Tables 1 and 2 around here *** 137
Frequency-current (f-I) relation 138
We create the frequency-current (f-I) curves by injecting a noisy staircase current into 139
either compartment and calculating the somatic firing rate for each current step. The noisy input 140
current was an Ornstein-Uhlenbeck process (Larkum et al., 2004): 141
𝐼𝑖𝑛𝑗𝑖 (𝑡 + 𝑑𝑡) = 𝐼𝑖𝑛𝑗
𝑖 (𝑡) +𝜇(𝑡) − 𝐼𝑖𝑛𝑗
𝑖 (𝑡)
𝜏𝑑𝑡 + 𝜎𝑖𝐺𝑡√
2𝑑𝑡
𝜏 (5)
where 𝐼𝑖𝑛𝑗𝑖 (𝑡) is the injected current at the i-th compartment with mean 𝜇(𝑡), compartment-142
dependent standard deviation 𝜎𝑖 , and time correlation length 𝜏. 𝐺𝑡 is a random number generated 143
at each time point from a Gaussian distribution with mean = 0 and standard deviation = 1. We set 144
𝜏 = 3 𝑚𝑠 as in the experimental study (Larkum et al., 2004), and 𝑑𝑡, the time increment, equal to 145
the integration time step. The mean 𝜇(𝑡) increased over time between 0.2 and 0.75nA as a staircase 146
function with steps of 𝜇(𝑡) = 0.05𝑛𝐴 every 2 s. 147
*** Please insert Figure 1 around here *** 148
Modeling a population of L5-PCs 149
In addition to reproducing all main features of PC reported from intracellular recording 150
studies, we validated its usefulness to model large-scale extracellular electric potentials (e.g., LFP) 151
generated by cortical microcircuits. To that end, we simulated a neocortical column comprised of 152
1,000 L5-PCs. For now, they were not connected to each other. Nevertheless, this approach 153
allowed us to determine the transmembrane ionic current densities (active/returning) and laminar 154
LFP associated with synchronized apical-dendritic L5-PC Ca2+-spikes. The laminar LFPs and CSD 155
patterns were compared with those obtained by Suzuki and Larkum (2017). 156
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Calculating the LFPs 157
We calculate the LFP from the transmembrane currents generated by a collection of 158
neurons using the point-source approximation (Holt and Koch, 1999), which assumes that the 159
transmembrane currents through a compartment can be approximated as a single monopolar 160
source/sink placed in an extracellular medium at the center of the compartment. To compute the 161
transmembrane currents, we divided each compartment into regions (Figure 1B). This approach 162
permits the spatial separation of active ionic and passive returning (i.e., capacitive and leak) 163
currents. The basal-dendritic/somatic compartment was modeled by three regions: the basal 164
dendrites, the axon hillock, and the soma-oblique dendrites. The apical-dendrite/trunk 165
compartment was modeled by two regions: the distal trunk (including the main bifurcation point), 166
and the tufted apical-dendrites. Each region was represented by a single monopolar current 167
source/sink. The ionic and capacitive/leak currents are distributed between these regions as follow: 168
𝐼1
𝑖 = (1 − 𝛼𝐾𝑑𝑟) ∙ 𝐼𝐾𝑑𝑟 + (1 − 𝛼𝐶) ∙ (𝐼𝐶𝑠 + 𝐼𝑙
𝑠) (6)
𝐼2
𝑖 = 𝐼𝑁𝑎 (7)
𝐼3
𝑖 = 𝛼𝐾𝑑𝑟 ∙ 𝐼𝐾𝑑𝑟 + 𝛼𝐶 ∙ (𝐼𝐶𝑠 + 𝐼𝑙
𝑠) (8)
𝐼4
𝑖 = 𝐼𝐶𝑎𝐿 + 𝛼𝐾𝑠 ∙ 𝐼𝐾𝑠 (9)
𝐼5
𝑖 = 𝐼ℎ + 𝐼𝑁𝑎𝑝 + IM + (1 − 𝛼𝐾𝑠) ∙ 𝐼𝐾𝑠 + (𝐼𝐶𝑑 + 𝐼𝑙
𝑑) (10)
where I1i , I2
i , I3i , I4
i , and I5i are the total transmembrane currents (Figure 1B) of the basal 169
dendrites (1), axon hillock (2), top soma-oblique dendrites (3), distal trunk (4), and the apical-170
dendrite (5) regions, respectively. ICs and IC
d are the somatic and dendritic capacitive currents, 171
respectively; and are equal to 𝐶𝑚{𝑠,𝑑}
𝑑𝑉{𝑠,𝑑}/𝑑𝑡. The distribution of ionic currents in these five 172
regions was determined by taking into consideration the following physiological/morphological 173
characteristics. 174
First, because of their extensive surface area, returning (i.e., capacitive and leak) currents 175
were distributed only in dendrites. The scaling factor 𝛼𝐾𝑑𝑟, 𝛼𝐶, and 𝛼𝐾𝑠 were adjusted to reproduce 176
the CSD patterns reported by Suzuki and Larkum (2017). We separated the somatic capacitive and 177
the 𝐼𝐾𝑑𝑟 currents into their contribution by the basal and top soma-oblique dendrites. These regions 178
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possess a bigger area and a combined higher density of 𝐼𝐾𝑑𝑟 channels than the axon hillock 179
(Ramaswamy and Markram, 2015). In the axon hillock, we included only the Na+ current because 180
its density in this area is at least 50-fold higher than at proximal dendrites (Ramaswamy and 181
Markram, 2015). The 𝐼𝐶𝑎𝐿 and 𝐼𝐾𝑠 currents were incorporated in the main bifurcation point of the 182
trunk since this region is the Ca2+-spike excitability zone (Larkum et al., 1999b). The Ih current 183
was added to the apical dendrite compartment because of its high density in this region (Kole et 184
al., 2006) and critical influence on synaptically evoked activity in the distal apical dendritic arbor 185
(Harnett et al., 2015). The 𝐼𝑁𝑎𝑝 (Schwindt and Crill, 1995) and IM (Hay et al., 2011) currents were 186
also included in this area because of their role in the amplification/attenuation of synaptic currents 187
in the distal apical-dendrites. Finally, the capacitive current was added into this region since the 188
distal dendritic arbor covers a greater area than the Ca2+-spike excitability zone (Ramaswamy and 189
Markram, 2015). 190
We compute the LFPs at 16 equally spaced vertically aligned points to simulate the linear 191
microelectrode array (Michigan probe) used by Suzuki and Larkum (2017). As in their study, the 192
inter-electrode distance (ℎ) was 100 𝜇𝑚. Motivated by their stimulation protocol with the right-193
angled prism, we consider that the linear probe was located at the center of a cylindrical neocortical 194
column of 3 𝑚𝑚 in diameter, and with constant and isotropic electrical conductivity 𝜎 =195
0.323 𝑆/𝑚 (i.e., average across layers from Goto et al., (2010)) (Figure 1C). Given the maximal 196
current produced by individual PCs, 1,000 L5-PCs were required to generate CSD amplitudes in 197
the range reported by Suzuki and Larkum (2017). The electric potential at electrode position 𝑧𝑒𝑗 is 198
given by (Nicholson and Llinas, 1971): 199
𝜙(𝑧𝑒𝑗) =
ℎ
2𝜎∑ ∑ (√(𝑧𝑒
𝑗− 𝑧𝑛
𝑖 )2
+ (𝑥𝑛𝑖 )
2+ (𝑦𝑛
𝑖 )2
− |𝑧𝑒𝑗
− 𝑧𝑛𝑖 |) ∙
𝐼𝑛𝑖 (𝑡)
𝑉
𝑁𝑠
𝑛=1
𝑁𝑛
𝑖=1
(11)
where 𝐼𝑛i (𝑡) is the transmembrane current generated by the point-source 𝑛 of the neuron 𝑖; 200
𝑥𝑛𝑖 , 𝑦𝑛
𝑖 , and 𝑧𝑛𝑖 are the coordinates of the point-source 𝑛 of the network neuron 𝑖, and 𝑉 is the 201
volume of the cortical column. 𝑁𝑛 = 1,000 and 𝑁𝑠 = 5 represent the total number of neurons in 202
the network and the total number of regions in each neuron, respectively. The (𝑥𝑛𝑖 , 𝑦𝑛
𝑖 ) coordinates 203
of the neurons in the simulated neocortical column were generated randomly from a uniform 204
distribution. The 𝑧𝑛𝑖 coordinate of the axon hillock point-source/sink of the network neurons was 205
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also generated randomly from a uniform distribution with values between 1.025 mm and 1.450 206
mm (below the pia matter, Suzuki and Larkum (2017)). The location of the basal-dendrite, trunk 207
main bifurcation point and apical-dendrite point-sources were calculated relative to the location of 208
the neurons’ axon hillock. The basal dendrites point-source was always 0.15 mm bellow the axon 209
hillock, the main bifurcation point of the trunk was always 0.89 mm above the axon hillock 210
(Ledergerber and Larkum, 2010, Figure 12), and the apical dendrite point-source was 0.15 mm 211
above the trunk main bifurcation point. The position of the top-soma oblique dendrites, 212
representing part of the somatic returning currents, was generated randomly with values between 213
0.7 and 1 mm from the cortical surface. The proposed distribution of point-sources for dendrites 214
was inspired by morphological data of L5-PCs (Mohan et al., 2015). Wiener noise 𝑔𝑘𝑑𝑊𝑘 was 215
added to equations (1) and (2) to instantiate variability in the timing of L5-PC Na+-APs and Ca2+-216
spikes, with 𝑔𝑠 = 0.05 and 𝑔𝑑 = 0.025. 217
Current source density (CSD) analysis 218
We estimated the CSD patterns evoked by the simulated LFPs using the spline inverse 219
CSD method (spline iCSD) (Pettersen et al., 2006). The iCSD methods are based on the inversion 220
of the solutions of the electrostatics forward problem and assume cylindrical confined and 221
symmetric CSDs. Specifically, the spline iCSD method assumes a continuously varying CSD 222
along the recording electrodes, which is calculated by interpolating a set of cubic splines, requiring 223
the CSD and its first and second derivatives in the vertical direction to be continuous (Pettersen et 224
al., 2006). It also considers a homogeneous disc distribution in the in-plane (𝑥, 𝑦) directions. In 225
agreement with pthe revious section, a homogeneous and isotropic volume conductor with 226
extracellular conductivity of 𝜎 = 0.323 𝑆/𝑚 (Goto et al., 2010) was used. Based on L5-PC density 227
and the CSD peak amplitudes in Suzuki and Larkum (2017), the diameter of the cylindrical source 228
model was set to 3 𝑚𝑚. The estimated CSD based on the simulated LFPs were convolved with a 229
Gaussian filter of 𝜎 = 0.1 𝑚𝑚 to produce a spatially smoothed CSD estimate. 230
Simulations and code accessibility 231
Simulations were performed in MATLAB (R2018b, MathWorks) with custom-written 232
scripts. The model equations are solved using the SDETools toolbox for the numerical solution of 233
stochastic differential equations (https://github.com/horchler/SDETools), with a time-step of 1 𝜇𝑠. 234
All simulation parameters are listed in Table 1 with ionic channel kinetics in Table 2. To calculate 235
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the CSD, we created customized scripts that use the functions provided in the CSDplotter toolbox 236
(https://github.com/espenhgn/CSDplotter), which implements the iCSD methods described in 237
Pettersen et al., (2006). The MATLAB scripts of the model implementation as well as for the LFPs 238
and CSD calculations are publicly available at (https://github.com/beaherrera/2-239
compartments_L5-PC_model). 240
Results 241
Model testing approach 242
Traditionally, parameter estimation of L5-PC biophysical models is performed using 243
quantitative strategies aimed at numerically minimizing model prediction errors while reproducing 244
transmembrane potential traces in specific experimental paradigms. In some cases, the data are 245
used to fit channel kinetics (Rapp et al., 1996), while in others (Hay et al., 2011; Bahl et al., 2012; 246
Almog and Korngreen, 2014; Chua et al., 2015; Mäki-Marttunen et al., 2018) conductance ranges 247
are fitted with generic optimization methods, based on known channel kinetics. However, such a 248
quantitative approach is very challenging if biophysical models are used to simultaneously fit data 249
from multiple experimental paradigms. In such cases, a qualitative trial/error approach based on 250
electrophysiological knowledge about the effect that each ion channel produces on the data is more 251
effective (Schaefer et al., 2003; Larkum et al., 2004; Yi et al., 2017). We will use the qualitative 252
trial/error approach as our goal is to satisfice qualitatively and not satisfy quantitatively six 253
different properties of L5-PCs (Table 3), which were reported using a variety of experimental 254
paradigms. We also employed previously known channel kinetics. The rationale used to determine 255
ionic distributions and conductances is now explained. 256
Ion channels for each compartment were selected based on experimental findings and 257
modeling studies. In the soma, we included the classic Na+ and K+ delayed rectifier channels to 258
generate the APs (Hodgkin and Huxley, 1952). Previous studies (Lytton and Sejnowski, 1991; 259
Larkum et al., 2004; Hay et al., 2011; Mäki-Marttunen et al., 2018) reported the need for the after-260
hyperpolarization (AHP) current to reproduce the f-I relationship shown experimentally by the L5-261
PCs. However, as in Bahl et al., (2012), this current was not needed to explain the f-I relationship. 262
On the other hand, the 𝐼𝐶𝑎𝐿 (Almog and Korngreen, 2009; Pérez-Garci et al., 2013), the 𝐼𝑁𝑎𝑝 263
(Schwindt and Crill, 1995; Crill, 1996) and the 𝐼𝐾𝑠 (Harnett et al., 2013) currents were inserted in 264
the dendritic compartment to generate the characteristic shape of dendritic Ca2+-spikes and in 265
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agreement with experimental data. 𝐼𝐶𝑎𝐿 defined the amplitude and velocity of the initial 266
depolarization phase. 𝐼𝑁𝑎𝑝 was responsible for the long plateau-like depolarization characteristic 267
of these spikes, and also essential for the CF effect. 𝐼𝐾𝑠 defined both the duration of the 268
repolarization phase and the amplitude of the Ca2+-spike’s after-hyperpolarization phase. The Ih 269
current (Kole et al., 2006) was included in the apical-dendrite/trunk compartment to account for 270
the increase in the resting potential reported in the dendrites of these neurons (Berger et al., 2001). 271
Moreover, this current plays a significant role in the BAC firing modulation and the synaptic 272
integration, as well as in the reported changes in both the CF for Ca2+-spike generation (Berger et 273
al., 2003) and the CSD pattern evoked by dendritic Ca2+-spikes (Suzuki and Larkum, 2017). 274
Finally, the M-current was needed for the spike repolarization phase when staircase input currents 275
were applied to the apical dendrites. Without this current, the dendritic membrane potential could 276
not complete the repolarization phase. The voltage dependence of the channel kinetics at the 277
apical-dendrite/trunk compartment was shifted by +8 mV to account for the shift in the resting 278
membrane potential. 279
Henceforth, we tested our L5-PC biophysical model in two steps. We first validate the 280
minimal model by reproducing all known Ca2+-dependent synaptic facilitation features. We next 281
assess the capabilities of the model to reproduce the large-scale Ca2+-spike dependent LFPs 282
associated with the synchronized activation of a population of L5-PCs in a neocortical column 283
responding to supra-CF somatic stimulation. 284
Validation of the model 285
Frequency-current (f-I) relationship 286
We first investigated whether our model predicts the f-I relationship previously reported 287
for L5-PCs when either the soma or the distal-trunk region is stimulated (Figure 2A, Larkum et 288
al., (2004)). We injected into the soma or the distal-trunk, a staircase incrementing noisy input 289
current generated using the Ornstein–Uhlenbeck method (see Materials and Methods), with 290
standard deviation 𝜎 = 0.2 𝑛𝐴, or 𝜎 = 0.09 𝑛𝐴, respectively. Figure 2B shows the somatic AP 291
response (blue, top panel) to the somatic input current (blue, second panel). Then, the mean 292
somatic AP frequency was computed for each current step. Mean and standard errors of the mean 293
(SEM) over 50 trials were estimated (Figure 2C, blue). Overall, the model predicted a linear f-I 294
relationship for the somatic input current (dashed blue line, goodness-of-fitting R2 = 0.959) that 295
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fell within the range of two experimentally reported studies for L5 PCs (Figure 2C, black 296
traces/shadow) (Larkum et al., 2004; Bahl et al., 2012). Figure 2B shows the somatic AP response 297
(blue, third panel) to the dendritic input current (red, bottom panel), and Figure 2C compares 298
observed with simulated values. The model also predicted a linear f-I relationship for dendritic 299
input current (dashed red line, goodness-of-fitting R2 = 1.00). In agreement with experimental data 300
(Larkum et al., 2004), current injections at the trunk must be ~300 pA larger than those needed at 301
the soma to produce the same AP frequency in these L5-PCs. This effect was quantified using 302
parameter ΔI (Figure 2D), which was calculated for all somatic AP rates from simulated (N = 6, 303
ΔI = 0.3142 ± 0.0140 nA) and experimental (N = 6, ΔI = 0.3333 ± 0.0258 nA) data. This difference 304
was statistically insignificant (t(5) = 2.0789, p = 0.0922, two-tailed paired t-test) demonstrating 305
the model represents adequately the experimental f-I relationships. However, our model predicted 306
a threshold for somatic AP initiation of ~0.35 𝑛𝐴 at trunk current injection sites, which was smaller 307
than that of ~0.5 𝑛𝐴 reported experimentally. This negative finding could be explained by the 308
difference in the injection site along the trunk of the actual L5-PCs used in the experiments 309
(Larkum et al., 2004). Current injections at sites relatively distant to the trunk bifurcation site will 310
require larger amplitudes as the density of CaL channels might be lower. In our model, the 311
mimicked injection was consistently performed at the level of that bifurcation lowering the 312
threshold required to achieve somatic AP firing. 313
*** Please insert Figure 2 around here *** 314
Back-propagating AP activated Ca2+-spike (BAC) firing 315
Next, we examined how the L5-PC biophysical model responds and integrates inputs into 316
the distal-trunk and soma (Figure 3A) at different times. Firstly, we stimulated the distal-trunk 317
with a subthreshold current generated from a double exponential function of the form 318
(1 − exp(−1/𝜏2)) ∙ exp (−1/𝜏2) with 𝜏1 = 2 𝑚𝑠 and 𝜏2 = 10 𝑚𝑠, and an amplitude of 0.29 𝑛𝐴. 319
In agreement with experimental studies (Larkum et al., 1999b; Schaefer et al., 2003), only a small 320
somatic and apical-dendritic/trunk depolarization were evoked by this current injection (Figure 321
3B). Second, we injected a threshold current pulse (duration: 5 𝑚𝑠, amplitude: 1 𝑛𝐴) into the 322
soma, which elicited an AP that propagated back to the apical-dendrite/trunk compartment creating 323
a dendritic depolarization but no Ca2+-spike (Figure 3). Third, we tested the model response when 324
both stimuli were combined. We applied the somatic current pulse and 1 𝑚𝑠 later the subthreshold 325
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current at the trunk. This resulted in the generation of an AP, a dendritic Ca2+-spike, and another 326
somatic AP following the onset of the dendritic Ca2+-spike (Figure 3D). We could also evoke 327
dendritic Ca2+ spikes by supra-threshold current injections to the trunk (Figure 3E). 328
*** Please insert Figure 3 around here*** 329
Critical frequency (CF) for Ca2+-spike generation 330
We next investigated the influence of the frequency of short somatic current stimulation 331
on Ca2+-spikes occurrence. To that end, we simulated the soma stimulation with trains of brief 332
supra-threshold pulses (2 𝑚𝑠) at different frequencies eliciting trains of somatic APs. As 333
previously reported (Larkum et al., 1999a; Berger et al., 2003), only AP trains above a CF (149 𝐻𝑧 334
in the model) evoked Ca2+-spikes. Figure 4A illustrates the somatic and apical-dendrite/trunk 335
responses to somatic stimulation below and at the CF. Figure 4B shows the intracellular Ca2+ 336
concentration dynamics for both stimulation paradigms, which resemble experimental data 337
(Larkum et al., 1999a). 338
*** Please insert Figure 4 around here *** 339
We also studied how the CF varied with the presence or absence of the Ih current in the 340
distal apical dendrites. We simulated a L5-PC without Ih current at the apical-dendrite/trunk 341
compartment responding to the same trains of supra-threshold currents at the soma with different 342
frequencies. To quantify the CF, we measured the area below the dendritic voltage traces and 343
plotted them as a function of AP frequency. When the Ih current was blocked relative to present, 344
the CF was lower by about 40 Hz (Figure 4C). Furthermore, we compared the CF values with and 345
without the Ih current predicted by our model with those predicted by experimental data from 346
eleven L5-PCs (Berger et al., 2003) (Figure 4D). In both the observed and simulated data, the CF 347
is reduced at least by 30-40 Hz when the Ih current is blocked. The CFs predicted by our model are 348
slightly higher than the mean observed CFs, but they fell within the observed range. 349
Reproducing Ca2+-spike dependent local field potentials 350
To examine the capabilities of this minimal L5-PC model, we tested whether non-synaptic 351
events such as Ca2+-spikes can be detected in the evoked LFPs as reported by Suzuki and Larkum, 352
(2017). To that end, we simulated a collection of 1,000 model L5-PCs (Figure 1C). In the 353
experimental paradigm, simultaneous stimulation of the soma of L5 PCs was achieved using an 354
optogenetic approach (Suzuki and Larkum, 2017); hence, no synaptic connections were considered 355
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in our simulations. The simulated L5-PCs were stimulated with a noisy, 20 ms duration, current 356
pulse with a mean amplitude that generates AP trains at a frequency above the CF. The mean input 357
current was strong enough to generate somatic AP trains and therefore evoked dendritic Ca2+-358
spikes (Figure 5A). Figure 5B shows the raster plots and associated post-stimulus time histograms 359
of 100 randomly selected L5-PCs (top), with the timing for typical Na+-APs and Ca2+-spikes. After 360
the somatic stimulation ceased, somatic Na+-APs were only elicited because of the non-linear 361
changes in the somatic-dendritic gain of these cells. 362
Figure 5C illustrates the averaged LFPs evoked by optogenetic stimulation of the collection 363
over 10 trials. We observed an early sink between 1.0-1.3 mm below the pia matter, which was 364
accompanied by two sources, one stronger between 0.7-0.9 mm and another weaker between 1.4-365
1.6 mm. According to our model, the sink was caused by large 𝐼𝑁𝑎 inward currents at the level of 366
the axon hillock due to the optogenetically induced APs. The two sources were caused by a 367
combination of 𝐼𝐾𝑑𝑟 and the returning capacitive/leak outward currents through the top soma-368
oblique dendrites and the basal dendrites. The relative amplitudes of these two sources can be 369
adjusted by means of parameters 𝛼𝐾𝑑𝑟 and 𝛼𝐶. To create Figure 5, these parameters were set to 370
0.5 and 1/3, respectively. We also observed a 20-30 ms delayed sink between 0.3-0.6 mm below 371
the pia matter, which was accompanied by a very superficial (0.1-0.2 mm) source, also delayed. 372
This late sink appeared during the same interval in which the collection of L5 PCs generated more 373
Ca2+-spikes (Figure 5B bottom). Hence, we believe it was caused by the 𝐼𝐶𝑎𝐿 inward current. 374
According to our model, the superficial sources resulted from a combination of 𝐼𝑀, 𝐼𝐾𝑠, and 375
the returning capacitive/leak outward currents through the apical-dendrites. Because of its reversal 376
potential, the cation current 𝐼ℎ could be either a source or a sink during a Ca2+-spike at a very 377
superficial level. The relative amplitude of this delayed sink-source was adjusted using parameter 378
𝛼𝐾𝑠 = 1 to reproduce a similar CSD pattern as that reported by Suzuki and Larkum (2017). The 379
CSD analysis clearly revealed the presence of such a sink/source current density distribution 380
(Figure 5D, right color map panel (Ih) and expanded plot, respectively). Since we did not consider 381
synaptic connections between the L5-PCs, the above results suggest that the late sink is associated 382
with the dendritic Ca2+-spikes. 383
*** Please insert Figure 5 around here *** 384
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Finally, we investigated the influence of the Ih current on the source-sink pattern generated 385
by the dendritic Ca2+-spikes. We repeated the simulations, but now without the Ih current in the 386
apical-dendritic compartment (Figure 5D). In agreement with the experimental data (Suzuki and 387
Larkum, 2017), we found that the amplitude of the delayed sink in layer 2/3 is significantly 388
increased by blocking the Ih current (p = 0.0089, Wilcoxon signed-rank test, N = 10 trials). Since 389
the superficial source in Suzuki and Larkum (2017) was very close to the pia boundary, we believe 390
the iCSD method used by the authors might have misestimated this source. Therefore, we did not 391
compare the experimental effect of blocking Ih on that superficial sources with that predicted by 392
our model. The cation current Ih was too small in amplitude to produce any detectable change in 393
the CSD when blocked. However, this current was crucial to produce a shifted resting membrane 394
potential of +10 mV (Figure 1C) in the apical-dendrite/trunk compartment, which disappeared 395
when Ih was blocked. As the trunk resting membrane potential became more negative, the effect 396
of 𝐼𝐶𝑎𝐿 was larger between 25-35 ms after stimulation (Figure 5D), producing a more intense 397
delayed sink during Ca2+-spiking at the level of the L5-PC trunk. 398
Discussion 399
Synaptic integration in apical dendrites of L5-PCs is facilitated by several unique 400
characteristics of these neurons: (i) the f-I curves with differentiated sensitivity for the soma and 401
distal trunk, (ii) the BAC firing-based amplification of coincident apical-dendritic inputs, (iii) the 402
CF effect for eliciting Ca2+-spikes in the distal trunk, and (iv) its related increases in intracellular 403
[𝐶𝑎2+]𝑖 in apical-dendrites strengthening NMDA synaptic efficacy. Biophysical models with 404
different level of complexity have been proposed to account for single L5-PC features (Rapp et 405
al., 1996; Larkum et al., 2004; Chua et al., 2015; Yi et al., 2017) or combinations of features 406
(Schaefer et al., 2003; Hay et al., 2011; Bahl et al., 2012; Almog and Korngreen, 2014; Mäki-407
Marttunen et al., 2018). Models explaining combinations of features require many compartments 408
to capture realistic L5-PC morphology and more than four ionic channels per compartment, which 409
substantially increase the computational cost and time (Table 3). Consequently, using these models 410
in large-scale simulations of collections of L5-PCs requires special computational resources and 411
extensive time. Moreover, fitting them to large-scale electrophysiological data (e.g., LFP and 412
EEG) will be challenging, limiting interpretability and applications to other research areas. Also, 413
no previous model has accounted for the shift in the CF due to the influence of Ih, explained the 414
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CSD patterns associated with dendritic Ca2+-spikes evoked by somatic stimulation of PCs above 415
the CF, or the effect on these patterns of blocking Ih (Suzuki and Larkum, 2017). We proposed a 416
2-compartment model of L5-PCs with just seven ion channels that explain qualitatively all 417
abovementioned features. 418
*** Please insert Table 3 around here *** 419
Layer 2/3 PCs vs. layer 5 PCs 420
Does our model account for characteristics of PCs in layer 2/3 (L2/3)? Even though they 421
share many electrophysiological properties, L2/3-PCs have distinct features that differentiate their 422
role in the cortical microcircuit (Larkum et al., 2007). Similar to L5-PCs, L2/3-PCs have 423
excitability zones in both the axon initial segment and the distal apical dendrites. These act as two 424
different functional compartments that allow these neurons to associate inputs coming to the distal 425
apical dendrites with those coming to the soma or basal dendrites. Large Ca2+ influx and 426
regenerative dendritic potentials are also evoked by back-propagating action potentials above a 427
CF. Moreover, as in L5-PCs (Pérez-Garci et al., 2006), GABAergic inhibitory inputs to the distal 428
apical dendrites cause long-lasting reduction of dendritic activity. However, unlike L5-PCs, L2/3-429
PCs do not show long plateau-like dendritic depolarizations. Consequently, brief dendritic spikes 430
have less influence on somatic AP output. In fact, isolated dendritic potentials in response to supra-431
threshold dendritic stimulation are more common than dendritic spikes coupled to a somatic AP. 432
Furthermore, though coincident inputs to both functional compartments reduce the threshold for 433
dendritic spike generation, stronger dendritic inputs are needed to evoke an extra somatic AP. In 434
addition, L2/3-PCs display little attenuation in the dendritic response to long current injections 435
suggesting a low density of Ih channels in the dendrites, described as sag by Larkum et al. (2007). 436
Functional Implications: Microcircuitry underlying cognitive control 437
Cognitive control involves the suppression of automatic or impulsive behavior for 438
successful goal-directed behavior. Some models of cognitive control formalize this function as the 439
co-activation of two conflicting action plans, which need to be resolved for correct performance 440
(Botvinick et al., 2001). Coincidence detection can also support error detection – a mismatch (or 441
conflict) between task goals and actual behavior – and prediction error – a mismatch between 442
expected and experienced outcomes (Alexander and Brown, 2011; Bastos et al., 2012; Cohen, 443
2014). Human and macaque electrophysiology experiments have characterized a negativity in 444
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scalp potentials associated with these cognitive functions (Gehring et al., 1993). Two components: 445
an N2 for conflict detection and the ERN for error detection. While the N2 and ERN are indices 446
of cognitive control, studying signal processing at the microcircuit level is essential to 447
understanding actual mechanisms (Cohen, 2017). Our biophysical model offers a powerful tool to 448
test different hypotheses and instantiate circuit models motivated by recent research sampling 449
neural spiking and LFP across frontal cortical layers (Chandrasekaran et al., 2017; Bastos et al., 450
2018; Sajad et al., 2019). 451
Recent models have proposed that conflict detection can be achieved by the detection of 452
coincident synaptic inputs in the medial frontal cortex (Alexander and Brown, 2011; Cohen, 2014; 453
Dembrow et al., 2015). L5-PCs can provide the neural substrate for the coincidence detection as 454
they have large dendritic trees that allow for integration across inputs from cognitive, limbic, and 455
motor structures (Huerta and Kaas, 1990; London and Häusser, 2005; Morecraft et al., 2012; Beul 456
and Hilgetag, 2015). Recently, we found that following errors in the stop-signal task, in a medial 457
frontal area, error-related neural spiking was first observed in putative pyramidal neurons in L5 458
and lower L3 (Sajad et al., 2019) concurrent with sinks in current in the superficial layers where 459
these neurons extend their dendrites (Sajad et al., 2017). Figure 6 diagrams our conjecture on the 460
role of L5-PCs in error detection in agranular cortex guided by the knowledge of the microcircuitry 461
and known anatomical connections. L5-PCs can detect the coincidence of an efferent copy of the 462
motor command from the mediodorsal thalamus and the task rule from prefrontal cortex (Sajad et 463
al., 2019). A mismatch between the two signals can result in spiking activity that can project 464
extrinsically to other structures (Barbas, 2015) and intrinsically to other neurons in the microcircuit 465
(Douglas et al., 1995; Haeusler and Maass, 2007; Kajikawa and Schroeder, 2011). L5-PCs are 466
densely interconnected with each other (not shown) resulting in rapid synchronous excitation of a 467
large number of L5-PCs upon receiving input currents (Hempel et al., 2000; Wang et al., 2006; 468
Morecraft et al., 2012). L5-PCs are also connected to inhibitory interneurons, which result in the 469
control of this excitation. Noteworthy, recent evidence suggests inhibitory neurons in agranular 470
cortex support more intra- than inter-laminar connections (Kätzel et al., 2011; Beul and Hilgetag, 471
2015). Hence, the inter-laminar inhibitory projections depicted in Figure 6 represent inhibitory 472
influences that are likely mediated by additional PCs and interneurons in L3 and L5 (not shown). 473
Previous studies have suggested that the patterns of interconnectivity of PCs and their 474
inhibition, particularly by the somatostatin-positive neurons, is important for controlled patterns 475
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of theta-band rhythmogenesis in the medial frontal cortex (Mainen and Sejnowski, 1996; Cohen, 476
2014), and its genesis has been associated with the role of the Ih current (Ulrich, 2002). Also, 477
intrinsic signal processing, involving inhibition of L5-PCs by interneurons in the upper layers has 478
been associated with the generation of gamma oscillations (Buzsáki and Wang, 2012; Bastos et 479
al., 2018). These oscillations result in scalp potential reflections of the gamma rhythm as well as 480
the theta rhythm, one the hallmarks of error and conflict detection (Tallon-Baudry and Bertrand, 481
1999; Cohen and Donner, 2013). 482
*** Please insert Figure 6 around here *** 483
While Figure 6 provides one explanation for signal flow within the microcircuit, it is far 484
from complete and relies on untested assumptions. For instance, the location where inputs to L5-485
PCs converge and the mechanism for how these signals are integrated at the biophysical level 486
remains technically challenging to study (Stuart and Spruston, 2015). Furthermore, the interaction 487
between L5-PCs and other neurons in the microcircuit remains unclear. The biophysical model 488
proposed in the current study provides an essential tool for further testing between and refining 489
competing hypotheses. Future work needs also the development of similar biophysical models for 490
L2/3-PCs and other neurons in the microcircuitry. 491
A necessary step towards understanding the origin of the ERN 492
The proposed model will be useful for another research goal of developing a forward model 493
of the ERN component. Clearly, the EEG arises from the activity of neurons in the brain tissue, 494
but the detailed relationship to activity within neocortex remains unclear (Riera et al., 2012; 495
Einevoll et al., 2013; Reimann et al., 2013). Recently, we have shown that error-related spiking 496
activity of neurons in the upper layers, but not lower layers of monkey supplementary eye field 497
predicts the magnitude of the ERN (Sajad et al., 2019). Also, recent work recording from single 498
neurons in humans have shown coupling between error neuron activity and intracranial EEG (Fu 499
et al., 2019). Somatic action potentials are unlikely to directly influence the EEG due to their 500
voltage dynamics; however, bursts of action potentials from a large population of neurons can 501
influence the EEG (Buzsáki et al., 2012). Furthermore, the large-scale EEG topography of Ca2+-502
spikes remains unknown (Suzuki and Larkum, 2017). The proposed biophysical model of L5-PC 503
can be used to directly examine how current flow resulting from neuronal activity within the 504
microcircuit, down to fine details of current dynamics, can result in LFP and scalp EEG (Riera et 505
al., 2012). Establishing a link between specific microcircuit motifs and fluctuations in the scalp 506
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potentials can render ERN more effective markers of specific cortical processes and stronger 507
diagnostic tools for patients with compromised cognitive control functions. 508
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Figure and Table Legends 692
Figure 1. Illustration of the biophysical model, LFP estimation and simulated neocortical column. 693
A – Equivalent circuit of the 2-compartment biophysical model. The first and second portions of 694
the circuit represent the basal-dendritic/somatic and apical-dendritic/trunk compartments, 695
respectively. The lengthy trunk is represented by the transfer resistance (𝑅𝑇) between the 696
compartments. Each ionic channel (k-th) is represented by an electromotive force 𝑬𝒌 (i.e., the ion 697
equilibrium potential) and a voltage-dependent conductance 𝒈𝒌 in parallel. B – Illustration of the 698
forward-modeling used for LFP estimation from the compartmental model of the L5-PCs. To 699
compute the transmembrane currents, the cell was divided into five current source/sink regions 700
(indicated by rectangles). The position of the point source/sink representing the compartment of a 701
neuron is given by the parameter 𝒓𝒏 = {𝑥𝑛, 𝑦𝑛, 𝑧𝑛}. The position of a representative electrode is 702
given by the parameter 𝒓𝒆 = {𝑥𝑒 , 𝑦𝑒 , 𝑧𝑒}. C – Diagram of the simulated cortical column formed by 703
a collection of 1,000 L5 PCs. The somas were distributed randomly in the tangential dimension of 704
layer 5. The mean (standard deviation) depth of the neurons was 1.04 (0.22) mm from the pia 705
matter. The simulated cortical column had a diameter of 3 mm and a total depth of 1.6 mm. As 706
previously reported (Berger et al., 2001), we used a resting membrane potential of -65 mV at the 707
somatic compartment, which drifted to -55 mV at the apical-dendritic/trunk compartment due to 708
the presence of the Ih current. 709
Figure 2. Frequency - Input (f-I) relationship. A – Micrograph of a L5-PC with recording locations 710
at the soma (blue) and distal trunk (red) indicated with diagram pipettes. B – Somatic AP responses 711
(1st and 3rd panels) to the staircase incremented noisy input current (2nd and 4th panels) injected 712
into the soma (blue) and distal trunk (red). C – Observed (black, Larkum et al., 2004; Bahl et al., 713
2012) AP firing frequency as a function of the mean input current. The range of observed values 714
is highlighted by a gray fill. Simulated mean and SEM spike rate over 50 trials for each current 715
step in the soma (blue) or distal trunk (red) compartment. Superimposed are observed (black 716
dashed) and simulated (blue and red dashed) linear regressions. D – Current differences (ΔI) 717
between the f-I curves for somatic and distal trunk stimulation to produce the same Na+-AP firing 718
frequency. No significant differences were found between the observed ΔI, numerically estimated 719
from Larkum et al., (2004) and that predicted by the model (t(5) = 2.0789, p = 0.0922, paired two-720
tailed t-test). 721
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Figure 3. Back-propagating AP activated Ca2+-spike firing. A – Micrograph of a L5-PC with 722
recording locations at the soma (blue) and distal trunk (red) indicated with a schematic pipette. B 723
– Simulated (left) and observed (right, Schaefer et al., (2003)) subthreshold current injected into 724
the apical dendrites creates only subthreshold somatic and dendritic depolarization. C – Simulated 725
and observed suprathreshold somatic current pulse elicits an AP that propagates back to the apical 726
dendrites creating a dendritic depolarization but no dendritic Ca2+-spike. D – Simulated and 727
observed combined somatic and tuft stimulation evokes an AP, a dendritic Ca2+-spike, and another 728
somatic AP following the onset of the dendritic Ca2+-spike. E – Simulated and observed 729
suprathreshold stimulation of distal apical dendrites evokes a dendritic Ca2+-spike. Scales in C are 730
common for all simulated (left) and observed (right) results. Red: apical-dendrite/trunks, Black: 731
basal-dendrites/soma, and Dashed Line: dendritic threshold. 732
Figure 4. Effect of somatic stimulation frequency on dendritic Ca2+-spike occurrence. A – A 733
simulated train of brief suprathreshold pulses at frequencies of 100 Hz (left) and 149 Hz (right) 734
(top) was injected into soma eliciting a train of APs (black, below). Only the 149 Hz train evoked 735
a dendritic Ca2+-spike (red, below). B – Intracellular dendritic Ca2+ concentration during somatic 736
stimulation at 100 Hz (left) and 149 Hz (right). Blue lines indicate the Ca2+ concentration at each 737
time instant of the dendritic voltage traces indicated in panel A. C – Integrated area below the 738
dendritic voltage traces as a function of the AP frequency with (blue) and without (black) blocking 739
the Ih current. CFs of 105 Hz and 149 Hz were obtained when Ih was present and absent, 740
respectively. D – Observed shift in CF after blocking Ih for eleven cells (black circles, numerically 741
estimated from Berger et al., (2003)) and simulated with the model (blue diamonds). 742
Figure 5. LFP and CSD derived from dendritic Ca2+-spikes in a collection of L5-PCs. A – Basal-743
dendritic/somatic (black, Vs) and apical-dendritic/trunk (red, Vd) simulated responses of a 744
collection of 1,000 L5-PCs to suprathreshold optogenetic stimulation above the CF. B – Raster 745
plots (top) and post-stimulus time histogram (bottom) of 100 randomly selected L5-PCs (top) 746
showing spike times of Na+-APs (black) and Ca2+-spikes (red). The total number of Na+-AP and 747
Ca2+-spike events every 5ms is shown (bar plots, bottom). C – LFPs evoked by the simulated 748
optogenetic stimulation of the collection of L5 PCs calculated on an array of 16 microelectrodes 749
(100 µm separation). Voltage traces at each depth are averaged over 10 simulated trials. The black 750
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28
rectangle indicates the delayed sink associated with the dendritic Ca2+-spike. D – CSD analysis of 751
the evoked LFPs averaged over 10 trials without (left) and with (right) the Ih current. The top right 752
panel expands the selected area to reveal the delayed sink associated with the Ca2+-spikes arising 753
earliest 0.4 mm below the pia matter. Middle right panel plots averaged 𝑰𝑪𝒂𝑳 current in the trunk 754
of the L5-PCs, showing an amplitude increase 25-35 ms after stimulation when Ih was blocked. 755
Lower right panel compares the average with SEM of the amplitude of this current sink with and 756
without blocking the Ih current, which was significantly different (p = 0.0089, Wilcoxon signed-757
rank test, N = 10 trials). The blue bar in all the plots indicates the time window for the optogenetic 758
stimulation. 759
Figure 6. Cortical microcircuit for coincidence detection underlying cognitive control. The 760
simplified diagram of circuitry embedding a L5-PC (blue) in agranular cortex with soma (triangle) 761
located in L5, dendrites that extend up to L1, and axons (blue arrows) that project both intrinsically, 762
innervating inhibitory neurons (red) and other pyramidal neurons (not shown) in the microcircuit, 763
and extrinsically, innervating other brain areas. This figure illustrates how dendritic dynamics can 764
contribute to an error signal. An efferent copy of a motor command is delivered through a 765
feedforward thalamic pathway, terminating on the L5-PC soma and apical dendrite. A task rule 766
signal from prefrontal cortex is delivered through a feedback pathway, terminating on the L5-PC 767
apical dendrites. The soma of a L5-PC (blue triangle) generates Na+-APs that propagate 768
intracortically to Martinotti cells (ovals) and other inhibitory interneurons (star). The Martinotti 769
cells terminate on the L5-PC apical dendrites, while the other interneuron terminates on the soma. 770
Note that because inhibitory neurons in agranular cortex largely make intra-laminar projections, 771
the inter-laminar inhibitory projections depicted here (dashed red lines) represent connections that 772
are likely mediated by additional PCs and interneurons in L3 and L5 (not shown). The dynamics 773
of this connectivity induces Ca2+ spikes, which amplify the coincidence of the efferent copy and 774
the task rule to generate an error signal. These neuronal events are signaled by the generation of 775
theta band LFP from deeper layers and gamma band LFP from superficial layers (indicated by 776
labeled oscillations). 777
778
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Table 1: Parameters used for the simulations. The first column indicates the ionic channels per 779
compartment. The second and third columns show the maximum conductance and equilibrium 780
potential for each ionic channel, respectively. The exponents of the activation (x) and inactivation 781
gating are indicated in the fourth column. Electrotonic parameters (capacitances/resistances) are 782
also shown. 783
Table 2: The gating kinetics for each ionic channel. 784
Table 3: Summary of previous/current biophysical models used to describe the principal features 785
of L5-PCs. The first, second, third and fourth columns show the study, number of compartments, 786
number of ionic channels and the platform used to create the simulated data for each study. The 787
fifth column list the features that were explained by each study. The column “Ions” provides the 788
following information 𝑁𝑠: 𝑁𝑐 𝑐⁄ , where 𝑁𝑠 and 𝑁𝑐 are the number of ionic species considered and 789
the number of ionic channels per compartment, respectively. In some cases, the number of ionic 790
channels per compartment depends on the regions of the neuron considered. Acronyms: IF – 791
Integrated and Fire Model; S – Soma; AD – Apical Dendrites; BD – Basal Dendrites; T – Trunk; 792
and Ah – Axon Hillock. 793
794
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30
Table 1 795
𝐠𝐤 (𝛍𝐒) 𝐄𝐤 (𝐦𝐕) (x,y)
Somatic Compartment
𝑪𝒎𝒔 = 𝟎. 𝟐𝟔 𝒏𝑭, 𝑹𝒎
𝒔 = 𝟓𝟎 𝑴𝜴 (Larkum et al., 2004)
Na 18 50 (3,1)
Kdr 5 -85 (0,4)
Leak 1/Rms -31.5 (0,0)
Dendritic Compartment
𝑪𝒎𝒅 = 𝟎. 𝟏𝟐 𝒏𝑭, 𝑹𝒎
𝒅 = 𝟒𝟑 𝑴𝜴 (Larkum et al., 2004)
Nap 0.022 50 (3,1)
CaL 3.85 RT
zFln (
[Ca2+]o
[Ca2+]i)
(2,0)
h 0.865 -45 (1,0)
M 1 -85 (1,0)
Ks 28 -85 (2,1)
Leak 1/Rmd -48.1 (0,0)
𝐑𝐓 = 𝟔𝟓 𝐌𝛀 (Larkum et al., 2004)
[𝑪𝒂𝟐+]𝒐
= 𝟐 𝒎𝑴
796
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31
Table 2 797
Ionic Current Gating variables
𝑁𝑎
𝛼𝑚 = 0.1 ∙ (𝑉 + 40)/(1 − exp (−(𝑉 + 40)/10))
𝛽𝑚 = 4 ∙ 𝑒𝑥𝑝(−(𝑉 + 65)/18)
𝛼ℎ = 0.07 ∙ 𝑒𝑥𝑝(−(𝑉 + 65)/20)
𝛽ℎ = 1/1 + 𝑒𝑥𝑝(−(𝑉 + 35)/10)
𝐾𝑑𝑟 𝛼𝑚 = 0.01 ∙ (𝑉 + 55)/(1 − 𝑒𝑥𝑝(−(𝑉 + 55)/10 ))
𝛽𝑚 = 0.125 ∙ 𝑒𝑥𝑝(−(𝑉 + 65)/80)
𝑁𝑎𝑝
𝑚∞ = 1/(1 + exp(−(𝑉 + 52.6)/4.6))
𝛼𝑚 = 0.182 ∙ (𝑉 + 38)/(1 − exp(−(𝑉 + 38)/6))
𝛽𝑚 = (−0.124 ∙ (𝑉 + 38))/(1 − exp((𝑉 + 38)/6))
𝜏𝑚 =6
𝑇𝑎𝑑𝑗(𝛼𝑚 + 𝛽𝑚)
ℎ∞ = 1/(1 + exp(−(𝑉 + 52.6)/4.6))
𝛼ℎ = −2.88 × 10−6 ∙ (𝑉 + 17) ∙ (1 − exp((𝑉 + 17)/4.63))
𝛽ℎ = −6.94 × 10−6 ∙ (𝑉 + 64.4) ∙ (1 − exp((𝑉 + 64.4)/2.63))
𝜏ℎ =1
𝑇𝑎𝑑𝑗(𝛼ℎ + 𝛽ℎ)
𝐶𝑎𝐿 𝛼𝑚 = 1.6/(exp(−0.072 ∙ (𝑉 − 5)) + 1)
𝛽𝑚 = 0.02 ∙ (𝑉 + 8.69) ∙ (exp((𝑉 + 8.69)/5.36) − 1)
𝐾𝑠
𝑚∞ = 1/(1 + exp(−(𝑉 + 11)/12))
𝜏𝑚 = (1.25 + 175.03 ∙ exp(0.026(𝑉 + 10)))/𝑇𝑎𝑑𝑗 , for 𝑉 < −60
𝜏𝑚 = (1.25 + 13 ∙ exp(−0.026(𝑉 + 10)))/𝑇𝑎𝑑𝑗 , otherwise
ℎ∞ = 1/(1 + exp(−(𝑉 + 64)/11))
𝜏ℎ = 360 + (1010 + 24 ∙ (𝑉 + 65)) ∙ exp(−((𝑉 + 85)/48)2)
ℎ 𝛼𝑚 = 0.00643 ∙ (𝑉 + 154.9)/ (exp((𝑉 + 154.9)/11.9) − 1)
𝛽𝑚 = 0.00193 ∙ exp(𝑉/33.1)
𝑀
𝛼𝑚 = 0.0033 ∙ exp(0.1(𝑉 + 35))
𝛽𝑚 = 0.0033 ∙ exp (−0.1(𝑉 + 35))
𝜏𝑚 =1
𝑇𝑎𝑑𝑗(𝛼𝑚 + 𝛽𝑚)
𝑇𝑎𝑑𝑗 = 2.334−21
10
798
.CC-BY 4.0 International license(which was not certified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprintthis version posted January 30, 2020. . https://doi.org/10.1101/2020.01.29.925180doi: bioRxiv preprint
32
Table 3. 799
Study Geometry Compartment Ions Platform Features Explained
(Rapp et al.,
1996) Realistic >>1,000
2: 2/c
Total >>2,000 NEURON BAC firing
(Larkum et
al., 2004) Simplified 2
2: 1/c, +IF model
Total = 2
Not
reported f-I curves
(Schaefer et
al., 2003) Realistic 8
7: 6/c
Total = 48 NEURON
BAC firing
CF
(Hay et al.,
2011) Realistic 199
8: 8/c S(1),
6/c AD(198)
Total = 1,196
NEURON
BAC firing
CF
f-I curves
(Bahl et al.,
2012) Realistic 14
8: 1/c Ah(5), 5/c S(1),
1/c BD(1), 4/c AD(7)
Total = 39
NEURON BAC firing
f-I curves
(Almog and
Korngreen,
2014)
Realistic
Many
Compartments
50 𝜇𝑚 in length
8: 8/c
Total = Many NEURON
BAC firing
CF
(Chua et al.,
2015) Simplified 3
1: 1/c AD,
+ IF model
Total = 1
NEST BAC firing
(Yi et al.,
2017) Simplified 2
3: 2/c S(1), 1/c AD(1)
Total = 3 MATLAB f-I curves
(Mäki-
Marttunen
et al., 2018)
Realistic 4
10: 9/c S(1),
1/c BD(1), 7/c AD(2)
Total = 24
NEURON
BAC firing
f-I curves
[Ca2+]i
This study Minimum 2
7: 2/c BD/S(1),
5/c AD/T(1)
Total = 7
MATLAB
BAC firing
CF
f-I curves
[Ca2+]i
Ih effect
CSD maps
800
.CC-BY 4.0 International license(which was not certified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprintthis version posted January 30, 2020. . https://doi.org/10.1101/2020.01.29.925180doi: bioRxiv preprint
33
801
Figure 1 802
803
.CC-BY 4.0 International license(which was not certified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprintthis version posted January 30, 2020. . https://doi.org/10.1101/2020.01.29.925180doi: bioRxiv preprint
34
804
Figure 2 805
806
.CC-BY 4.0 International license(which was not certified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprintthis version posted January 30, 2020. . https://doi.org/10.1101/2020.01.29.925180doi: bioRxiv preprint
35
807
Figure 3 808
809
.CC-BY 4.0 International license(which was not certified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprintthis version posted January 30, 2020. . https://doi.org/10.1101/2020.01.29.925180doi: bioRxiv preprint
36
810
Figure 4 811
812
.CC-BY 4.0 International license(which was not certified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprintthis version posted January 30, 2020. . https://doi.org/10.1101/2020.01.29.925180doi: bioRxiv preprint
37
813
Figure 5 814
815
.CC-BY 4.0 International license(which was not certified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprintthis version posted January 30, 2020. . https://doi.org/10.1101/2020.01.29.925180doi: bioRxiv preprint
38
816
Figure 6 817
.CC-BY 4.0 International license(which was not certified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprintthis version posted January 30, 2020. . https://doi.org/10.1101/2020.01.29.925180doi: bioRxiv preprint